Propagation of Acoustical Wave in Finite Cylindrical Solid Bar surrounded by
Semi-Infinite Porous Media saturated with Fluid
Un-Son Ria , Un-Gyong Ana , Song-Jin Imb
arXiv:1511.03134v1 [cond-mat.mtrl-sci] 7 Nov 2015
b Department
a Department of Acoustics, and
of Optics, Faculty of Physics, Kim Il Sung University, Ryongnam-Dong, Taesong-District, Pyongyang, DPR Korea
Abstract
We established the propagation equation of acoustical wave in media with the solid/porous media cylindrical boundary
and obtained the analytic solution. We suggested the boundary condition on solid-porous media cylindrical boundary.
Based on that, we introduced the dispersion equation, and constructed the algorithm to perform numerical calculation
and analysis of dispersion equation.
Keywords: acoustic wave, porous media, fluid
1. Introduction
In the past, many research works have been performed in a fluid well surrounded by porous media. It is important
for the understanding and quantitative interpretation of acoustic and seismic measurements in hydrocarbon wells,
pipelines, as well as laboratory measurements to model wave propagation modes in cylindrical structures. For porous
materials these waves are affected by material permeability. These effects can be analyzed using Biots equations of
poroelasticity[1]. For a porous cylinder with open-pore boundary conditions on its surface this was first done by
Gardner[2], who derived the dispersion equation for extensional waves at low frequencies. For the full frequency
range and closed boundary conditions the dispersion in a fluid-saturated cylinder was studied by Berryman[3]. This
was done for the first few modes because conventional root finding becomes challenging for poroelastic media. An
alternative approach to modeling wave propagation in circular structures was recently introduced by Adamou and
Craster[4] based on the spectral method. Karpfingeret al. [5] extended the spectral method to axisymmetric waves for
arbitrary fluid and solid layers. In this paper we considered the solution of dispersion equation of modeling wave in
finite cylindrical solid bar surrounded by porous media saturated with fluid. Few literatures on sound propagation in
solid bar surrounded by porous media is available[6, 7]. Today, many applications such as logging by solid bar and
ultrasonic boring require theoretical studies on sound propagation in solid bar.
2. Wave equation and its solution in solid bar and porous media
In this paper, we assume that solid bar is homogenous axisymmetric one, porous media is the isotropic one whose
porosity is φ, and that linear dimension of porosity is much smaller than the wavelength. Displacement of wave is
given by [6, 7]
u = ∇ϕ + ∇ × ∇ × ψez
When the displacement oscillates harmonically, wave equation in solid bar may be written as
∇2 ϕ + k2p ϕ = 0
(1)
Email address: (Song-Jin Im)
Preprint submitted to Elsevier
November 11, 2015
∇2 ψ + k2p ψ = 0
(2)
where k2p = ω2 /c2p , c2p = (λ + 2µ)/ρ, k2s = ω2 /c2s , c2s = 2µ/ρ, and λ, µ are Lame’s constants and ρ is the density of bar,
∇2 is laplacian, ω is frequency of wave. In homogeneous and axial symmetrical bar the solutions of wave equation
are
(3)
ϕ = AI0 (kr r)eik0 z
ψ = BI0 (kr0 r)eik0 z
2
(4)
2
where kr = k02 − k2p , kr0 = k02 − k2s , k02 = ω2 /c2 , In is modified Bessel function of the first kind of nth order, and c is
velocity of acoustical wave in axial direction. Equation of motion in porous media saturated as fluid is as follows[8, 9]
− N[∇ × [∇ × u]] + (A + 2N)∇ · u + Q∇(∇ · U) = ρ11 ü + ρ12 Ü + b(u̇ − U̇)
(5)
Q∇(∇ · u) + R∇(∇ · U) = ρ12 ü + ρ22 Ü − b(u̇ − U̇)
(6)
where U and u are each displacement vector in solid and fluid, ρi j and b are mass combination coefficient and viscous
coefficient respectively, which are given follows[10].
b(ω) = β2 H1 (ω)
β2 H2 (ω)
ω
ρ12 (ω) = βρ f − ρ22 (ω)
ρ22 (ω) =
ρ11 (ω) = (1 − β)ρ s − ρ12 (ω)
Here
H1 (ω) + iH2 (ω) = 1/K(ω)
K(ω) is Darcy coefficient, which is expressed as following[11];
" #
iωρ 1/2
J2 ia η f
iβ
K(ω) =
" #
ωρ f
iωρ 1/2
J0 ia η f
ρ s and ρ f are densities of solid and fluid, a is characteristic radius, η is dynamic viscous coefficient of saturated fluid,
Jn is Bessel function of the first of order n, β is porosity. A, N, Q and R are elastic constants related with Lame’s
constants, which are given as;
m K + Ks K
(1 − β) 1 − β − K
s
Ks
Kf m 2
A=
− N
3
m + β Ks
1−β− K
K
K
s
m
1−β− K
Ks
Q=
Ks
m
1−β− K
Ks + β K f
R=
β2 − K s
Ks
m
1−β− K
Ks + β K f
N = µm
2
f
where K s , Km , K f are bulk moduli of solid, skeleton of porous medium and fluid respectively, µm is shear modulus of
skeleton of porous solid, which are given as follows
2
Km = (1 − β)ρ s Vl12 − 4V s1
/3
µm = (1 − β)ρ s v2s1
K f = ρ f Vl22
Vl1 , V s1 , Vl2 are velocities of longitudinal wave, shear wave in solid longitudinal wave in fluid of porous media
respectively. Let teh displacement vectors in solid, fluid within porous media be expressed as;
2
X
∇ϕ j + ∇ × ψ1 eϕ
(7)
ξ j (ω)∇ϕ j + χ(ω)∇ × ψ1 eϕ
(8)
u=
j=1
U=
2
X
j=1
where
2
ξ j (ω) = −
1
Rγ11 − Qγ12 (ω)
PR − Q
+
Vl2j (ω) Qγ22 (ω) − Rγ12 (ω) Qγ22 − Rγ12 (ω)
χ(ω) = −γ12 (ω)/γ22 (ω)
γii (ω) = ρii (ω) − ib(ω)/ω
γ12 (ω) = ρ12 + ib(ω)/ω
p = A + 2N
When acoustical field oscillates harmonically, we can introduce three Helmholtz’s equation as follows.
(9)
∇2 ϕ2 + k2p2 ϕ2 = 0
(10)
∇2 (ψ1 eϕ ) + k2s1 (ψ1 eϕ ) = 0
(11)
where
=ω
=ω
Vl1 , Vl2 are velocities, V s1 is velocity of rotational wave. Taking into account the
boundary condition, solutions of above equations are
k2p1
2
/Vl12 ,
∇2 ϕ1 + k2p1 ϕ1 = 0
2
K p2
2
2
/V s1
,
ϕ1 = A1 K0 (Kr1 R)eik0 z
ϕ2 = A2 K0 (Kr2 R)eik0 z
ψ1 = B1 K1 (Kr0 R)eik0 z
2
2
where kr1
= k02 − k2p1 , kr2
= k02 − k2p2 , kr02 = k02 − k2s1 .
3. solid-porous media cylindrical boundary condition
Cylindrical boundary condition of solid-porous media and dispersion equation are
(ur )I |r=a = (ur )II |r=a
(uz )I |r=a = (ur )II |r=a
(σrr )I |r=a = (σrr )II |r=a + σ|r=a
(12)
(σrz ) |r=a = (σrz ) |r=a
I
II
(Ur )II |r=a = (ur )|r=a
where superscripts I, II stand for solid and porous media. It is S = −βP[8], β is porosity, p is pressure in porous
fluid, ur , uz are displacements of solid in r and z directions, σrr , σrz are components of stress, Ur is displacement in
r-direction in porous fluid.
3
4. Dispersion equation
We calculated the components of displacement and stress from solutions of equations in bar and porous media.
The results are as follows.
(ur )I =Akr I1 (kr R) − Bik0 kr0 I1 (kr0 ),
2
(uz )I = − Aik0 I0 (kr R) − Bkr0 I0 (kr0 R),






2
kr


2
I
(λ + 2µ)kr − λk0 I0 (kr R) − 2µ I1 (kr R)
+
(σrr ) =A 




R




ik0 kr0
2
0
+ B −2µik0 kr0 I0 (kr0 R) + 2µ
I1 (kr R) e
R
(ur )II = − A1 kr1 K1 (kr1 v) − A2 kr2 K1 (kr2 R) − B1 ik0 K1 (kr0 R)
(Ur )II = − A1 ξkr1 K1 (kr1 R) − A2 ξ2 kr2 K1 (kr2 R) − B1 χik0 K1 (kr0 R)
(uz )II =A1 ik0 K0 (kr1 R) + A2 ik0 K0 (kr2 R) − B1 kr0 K0 (kr0 R)
#
"
2Nkr1
2
2
II
2
K1 (kr1 R) + (A + Qξ1 )(kr1 − k0 )K0 (kr1 R) +
(σrr ) =A1 2Nkr1 K0 (kr1 R) +
R
#
"
2Nkr2
2
2
2
+ A2 2Nkr2 K0 (kr2 R) +
K1 (kr2 R) + (A + Qξ2 )(kr2 − k0 )K0 (kr2 R) +
R
#
"
ik0
0
0
0
K1 (kr R)
+ B1 2N ik0 kr K0 (kr R) +
R
2
2
(σ)II =A1 (Q + Rξ1 )(kr1
− k02 )K0 (kr1 R) + A2 (Q + Rξ2 )(kr2
− k02 )K0 (kr2 R)
#
"
2Nkr1
2
2
II
2
K1 (kr1 R) + (A + Q + Qξ1 + R)(kr1 − k0 )K0 (kr1 R) +
(σrr ) + σ =A1 2Nkr1 K0 (kr1 R) +
R
"
#
2Nkr2
2
2
+ A2 2Nkr2
K0 (kr2 R) +
K1 (kr2 R) + (A + Q + Qξ2 + R)(kr2
− k02 )K0 (kr2 R) +
R
"
#
ik
0
+ B1 2N ik0 kr0 K0 (kr0 R) +
K1 (kr0 R)
R
(σrz )II = − A1 2Nik0 kr1 K1 (kr1 R) + A2 2Nik0 (−kr2 )K1 (kr2 R) + B1 N(kr02 + k02 )K1 (kr0 R)
σ
Q1 + Rξ1 2
Q + Rξ2 2
(P)II = − = −A1
(kr1 − k02 )K0 (kr1 R) − A2
(kr2 − k02 )K0 (kr2 R)
β
β
β
Substituting the displacements and stresses into boundary condition, we obtain following matrix equation,







m11
m21
m31
m41
m51
m12
m22
m32
m42
m52
m13
m23
m33
m43
m53
m14
m24
m34
m44
m54
4
m15
m25
m35
m45
m55

 
 
 
 
 
 
A
B
A1
A2
B1




 = 0,


(13)
where
m11 = kr I1 (kr a)
m21 = ik0 I0 (kr a)
2
kr
m31 = (λ + 2µ)kr − λk02 I0 (kr a) − 2µ I1 (kr a)
a
m41 = 2µik0 kr I1 (kr a)
m51 = 0
m12 = ik0 kr0 I1 (kr0 a)
2
m22 = −kr0 I0 (kr0 a)
2
m32 = 2µik0 − kr0 I0 (kr0 a) − 2µ
ik0 − kr0
I1 (kr0 a)
a
3
m42 = −µ(k02 kr0 + kr0 )I1 (kr0 a)
m52 = 0
m13 = kr1 K1 (kr1 a)
m23 = −ik0 K0 (kr1 a)
m33 = −2Nkr2 K0 (kr1 a) −
2Nkr1
2
K1 (kr1 a) − (A + Q + Qξ1 + Rξ1 )(kr1
− k02 )k0 (kr1 a)
a
m43 = 2Nik0 kr1 K1 (kr1 a)
m53 = (1 − ξ1 )kr1 K1 (kr1 a)
m14 = kr2 K1 (kr2 a)
m24 = −ik0 K0 (kr2 a)
2Nkr2
2
K1 (kr2 a) − (A + Q + Qξ2 + Kξ2 )(kr2
− k02 )K0 (kr2 a)
a
2
m34 = −2Nkr2
K0 (kr2 a) −
m44 = 2Nik0 kr2 K1 (kr2 a)
m54 = (1 − ξ2 )kr2 K1 (kr2 a)
m15 = ik0 K1 (kr0 a)
m25 = kr0 K0 (kr0 a)
m35 = −2N
ik0 kr0 K0 (kr0 a)
ik0
+
K1 (kr0 a)
a
!
m45 = −N(kr02 + k02 )K1 (kr0 a)
m55 = (1 − χ)ik0 K1 (kr0 a)
In order that matrix equation has a nontrivial solutions, the determinant of the coefficient must be zero. Thus, the
following dispersion equation must be satisfied;
m11 m12 m13 m14 m15 m21 m22 m23 m24 m25 m31 m32 m33 m34 m35 = 0
m41 m42 m43 m44 m45 m51 m52 m53 m54 m55 5. Numerical calculation and interpretation of dispersion equation
The dispersion equation is nonlinear equation of about 70th , which makes the analytic solution impossible, so
we must performed numerical evaluation. We used Newton method to do it. In case of the radius of bar a = 0.1m,
5
porosity φ = 0.19, viscous coefficient η = 0.01, density of bar ρ1 = 1500kg/m3 , velocity of longitudinal wave in bar
c p1 = 2450m/s, and velocity of tansversal wave in bar c s1 = 1500m/s, density of solid, longitudinal and transpersal
velocity in porous media ρ2 = 2650kg/m3 , c p2 = 3670m/s, c s2 = 2170m/s respectively, density of porous fluid
ρ f = 1000kg/m3 , velocity in porous fluid c p3 = 1500m/s, bulk modulus of porous solid k s = 3.79 × 1010 Pa, the
dispersion curve is shown in Figure 1. The dispersion curves have two groups. First group beginning from velocity
Figure 1: Dispersion curve of velocity
of transversal wave in porous solid approach to velocity of transversal wave in bar, and second group from velocity
of longitudinal wave in porous solid to velocity of longitudinal wave in bar. Imaginary parts of velocity of second
group are very large, but in contrast, those one of second group are very small and ignored. The first group of waves
correspond to Transversal reference wave modes of solid bar and second group to longitudinal reference wave modes.
In case of c s1 < c s2 , transversal reference wave modes satisfy condition of total reflection. But in contact of fluid of
porous media with the bar, the condition of total reflection is not satisfied, therefore part of wave may be lost. That’s
why imaginary part of wave number of undamped transversal reference wave appears in solid-to-solid boundary. And
the imaginary part of longitudinal reference wave of second group is large, so it can be shown that the condition
of total reflection is not satisfied by existence of refraction transversal wave when longitudinal reference waves are
reflected at boundary. Therefore, longitudinal reference wave modes become loss waves and transversal reference
waves are fundamental part in a wave field propagating in bar and longitudinal reference waves contribute little to
complete wave field because of its damping property.
6. Conculsion
In this paper we established the equation of acoustical wave propagation in media with the solid-porous media
cylindrical boundary and obtained its solution. We set solid-porous media cylindrical boundary condition to introduce
the dispersion equation, and performed numerical evaluation of nonlinear equation of 70th degree to obtain phase
velocity dispersion curves consisted of two groups, first of which corresponds to transversal reference wave mode,
and second of which to longitudinal reference wave mode. The wave of transversal reference wave mode is undamped
and that one of longitudinal reference wave mode is damped and ignored.
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6
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